2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/31109We show that Closest Substring, one of the most important problems in the field of biological sequence analysis, is W[1]-hard when parameterized by the number k of input strings (and remains so, even over a binary alphabet). This problem is therefore unlikely to be solvable in time O(f(k)\cdot n^{c}) for any function f of k and constant c independent of k. The problem can therefore be expected to be intractable, in any practical sense, for k>=3. Our result supports the intuition that Closest Substring is computationally much harder than the special case of Closest String, although both problems are NP-complete. We also prove W[1]-hardness for other parameterizations in the case of unbounded alphabet size. Our W[1]-hardness result for Closest Substring generalizes to Consensus Patterns, a problem of similar significance in computational biology.21 pages, 5 figures; an extended abstract of this paper was presented at the 19th International Symposium on Theoretical Aspects of Computer Science (STACS 2002), Springer-Verlag, LNCS 2285, pages 262--273, held in Juan-Les-Pins, France, March~14--16, 2002Computational ComplexityF.2;F.2.2;J.3text